After May's influential paper[1]}, the difference equations
gained wide popularity in all walks of science. They
were studied from
the viewpoint of nonlinear dynamics and chaos which was ubiquitous
in disciplines ranging from ecology to economics. Mathematicians
studied it as an interesting object in its right[2]}.
The studies on maps are computationally less intensive. In certain
cases, it was easier to track them analytically. Feigenbaum's study
of period-doubling bifurcations in logistic maps is an
example[3]}, [4]}.
Lower order difference equations can show the same phenomena
as higher-order differential equations. For example, we can observe chaos in
a logistic map that has a single variable while we need at least three first-order autonomous differential equations to observe chaos.
Insights
gained from studies in difference equations were often
(though not always) useful in studies in differential equations
and vice versa. From control problems to synchronization schemes,
many theoretical ideas have found applications in both difference
and differential equations[5]}. In this work,
we study fractional linear difference equations and coupled fractional linear difference equations. We show that the
analysis of linear fractional
difference equations has remarkable similarities with corresponding
fractional differential equations.
For coupled difference equations, the Jacobian, its eigenvalues
and eigenvectors play a central role. We show that similar concepts
can be very useful for coupled fractional difference equations.
